2.1. Spanwise flow simulation

Several representative wind turbine blade benchmarks are used in this study, ranging from early stage small-scale turbines to contemporary large-scale designs, as shown in figure 2. Furthermore, the comprehensive wind turbine experimental database ensures the universality of the 3-DGAR model.

2.1.1. The NREL Phase VI rotor

The NREL (National Renewable Energy Laboratory) Phase VI rotor is a widely adopted reference model in wind turbine aerodynamics research. As a representative kilowatt-scale wind turbine, it has a relatively low power rating and employs a single aerofoil section. Its twist distribution varies from a 14.29 $^\circ$ at 30 % to −2.69 $^\circ$ at 95 % spanwise position. Nevertheless, its geometric design philosophy continues to influence blade designs to the present day.

In this work, the parked and rotating experiments are selected, including the experimental sequences L and S with two-bladed configurations, a rigid hub and a cone angle of 0 $^\circ$ (Simms et al. Reference Simms, Schreck, Hand and Fingersh2001). The pressure sensors are distributed at a 30 %, 47 %, 63 %, 80 % and 95 % span to measure the pressure of the blade. In sequence L, the rotor is at a standstill at 0 $^\circ$ azimuth. The continuous ramp of the pitch was carried out with 0.180 degrees per second. For the NREL Phase VI blade in a stationary condition, the Reynolds number is approximately $0.75$ to $1.5 \times 10^6$ . The pitch increment step is 5 $^\circ$ , and then hold for 8 s under velocity 20 m s⁻¹. The rotational speed 72 rpm with different free-stream velocities $U_0$ was investigated in sequence S. In this study, the experimental data with 0 yaw degree and the wind speeds 5–20 m s⁻¹ are utilised to validate the 3-D geometric effects and the proposed 3-DGAR model.

Figure 2.

The experimental rotors and the corresponding aerofoils.

2.2. Framework of the 3-DGAR model

We begin by analysing the aerodynamic force coefficients of the standstill and rotation blades. Figures 4(a) and 4(b) illustrate the blade loads computed utilising BEMT under both stationary and rotating conditions. The classical BEMT framework is employed in this study, following the formulations presented in standard textbooks (Wang et al. Reference Wang, Zhong, Qian and Zhu2023). The Prandtl–Glauert tip-loss model is adopted to correct the induction factors in the momentum theory, while the Bak model is used to account for 3-D rotational effects. For broader applicability, our BEMT framework is also equipped with the Leishman–Beddoes dynamic stall model and a yaw correction model. Although many advanced yaw flow prediction methods are available (Bastankhah et al. Reference Bastankhah, Shapiro, Shamsoddin, Gayme and Meneveau2022; Heck et al. Reference Heck, Johlas and Howland2023; Li & Yang Reference Li and Yang2024; Li et al. Reference Li, Gao, Liu, Lu and Stevens2025), this study focuses on spanwise flow under uniform inflow and, thus, uses only the classical yaw correction model (Schepers Reference Schepers2012). The accuracy of this in-house BEMT framework has been thoroughly validated in our previous work (Shi et al. Reference Shi, Gao and Zhang2025).

Figure 3.

Definition of sectional angle of attack $\alpha _{{fs}}$ , inflow angle $\phi$ , pitch angle $\beta$ and local twist angle $\gamma$ for both (a) stationary and (b) rotating blades.

For the stationary blade, the free-stream angle of attack $\alpha _{{fs}}$ at a given blade section is determined by the sum of the blade pitch angle $\beta$ and the section twist angle $\gamma$ , under uniform and steady inflow in figure 3(a). Its incoming wind direction is perpendicular to the rotor plane with inflow angle $\phi = 90^\circ$ .

For a rotating blade, the angle of attack at various sections is affected by both the incoming wind velocity and the rotational speed, as shown in figure 3(b). Here, $W$ denotes the actual inflow velocity experienced by the blade section, accounting for the combined effects of the free-stream and rotational velocities. In the BEMT framework, the inflow angle $\phi$ is therefore determined by the free-stream velocity, the rotational velocity and the induced velocity factors $a$ and $a'$ , which are obtained through iterative coupling. Consequently, the angle of attack for a blade section can be formulated as

(2.1) \begin{align} \alpha _{{fs}}=\phi -(\beta +\gamma ). \end{align}

For non-rotating blades, the angle of attack depends only on pitch and twist, while the induced velocity ( $a$ and $a'$ ) in momentum theory is zero. Without additional corrections, the aerodynamic loads at different sections directly align with those at the angle of attack of the aerofoil profile. Thus, in figure 4(a) the 3-D loads calculated by BEMT are equivalent to the 2-D aerofoil loads (the red solid line). The aerodynamic force coefficients are defined based on the local chord length of the blade section.

Figure 4.

Comparison of experimental data for the NREL Phase VI rotor with predictions from conventional models, including the blade element theory and the rotational correction model (Bak model), under (a) parked and (b) rotational conditions.

Due to the prevalent blade design, which incorporates a positive twist near the root and a negative twist towards the tip, the most significant reduction in lift slope occurs at the inboard sections, with the effect gradually diminishing towards the tip as shown in figure 4(a). The lift slope at the 95 % tip section does not follow the aforementioned trend. However, BEMT neglects the spanwise flow and the geometric effects. It exhibits significant deviations across most of the sections, with particularly severe overpredictions at the root (30 % span) and tip (95 % span). Specifically, for blade tip load prediction, the relative error surpasses 30 %. Such load-estimation disparities may lead to unusual vibrations or excessive deformations at standstill.

For rotating blades, $\alpha _{{fs}}$ for different sections are solved iteratively by BEMT. This process involves interpolating from 2-D aerofoil polars and subsequently applying a rotational augmentation model to obtain the 3-D loads. As a result, figure 4(b) illustrates the blade loads obtained from BEMT after implementing the rotation effect corrections under different inflow angles. The three red lines represent the BEMT predictions at the spanwise sections corresponding to 30 %, 63 %, and 95 % respectively. The lift-curve slope for the rotating blade adheres to the same pattern as that of the stationary blade: it increases from the root to the mid-span, followed by a notable reduction in proximity to the tip. The Bak rotational models, incorporating the Prandtl–Glauert tip-loss correction, capture only the spanwise flow due to rotation and the associated load augmentation. However, they significantly overestimate the loads in the vicinity of the 95 % tip region.

From the perspective of aerodynamic loading, both stationary and rotating blades exhibit a similar trend: the lift slope ${\rm d}C_{\!L}/{\rm d}\alpha$ increases progressively from inboard to outboard sections, yet a decline in loading occurs near the tip, as shown in figure 5. Therefore, the lift-curve slope exhibits a trend inversely proportional to the spanwise position $r/R$ . Accordingly, a polynomial function of $ {1}/({r/R})$ is employed to fit the lift-curve slope, as indicated by the black line:

(2.2) \begin{align} \frac {{\rm d}C_{\!L}}{{\rm d}\alpha }=-0.001\frac {1}{r/R}-0.004\left(\frac {1}{r/R}\right)^{2}+0.108. \end{align}

Figure 5.

Variation of the lift slope ( ${\rm d}C_{\!L}/{\rm d}\alpha$ ) across different spanwise positions. The black solid line denotes the polynomial fitting function between the lift-curve slope and the spanwise position $r/R$ , while the dashed line indicates that the lift-curve slope near the blade tip deviates from the unified fitting function.

These spanwise variations of the lift-curve slope reveal two distinct mechanisms governing the aerodynamic loading from the root to the tip: (i) an alteration in the lift-curve slope from the root to the middle of the blade induced by the geometric effect, and (ii) an additional load reduction near the tip superimposed on pattern (i) corresponding to the tip flow.

Therefore, in order to model these two different flow patterns, the 3-DGAR model is represented as a classical and simple framework, as shown in figure 1:

(2.3) \begin{align} C_{\!L}(\alpha _{{fs}})=C_{l}(\alpha _{{eff}})+\Delta C_{\!L} .\end{align}

Accordingly, $C_l(\alpha _{{eff}})$ corresponds to the lift coefficient after accounting for downwash, while $\Delta C_L$ quantifies the additional load loss near the tip region. For the drag coefficient, the formulation is obtained by correcting with the effective angle of attack and subsequently adding the induced-drag contribution $C_{Di}$ . Notably, the classical Prandtl tip-loss correction pays less attention to the reduction in lift slope or lift coefficient in the tip region, as shown in figure 5. Therefore, $\Delta C_L$ is introduced to describe the direct variation of blade element load in the tip region.

Furthermore, the most important and challenging question is to model $\alpha _{{eff}}$ and $\Delta C_L$ under varying flow conditions and blade geometries. Theoretical models for each pattern will be developed subsequently, accompanied by investigations into the underlying flow mechanisms.

2.3. Downwash effect and the axial downwash velocity model

Figure 6.

Comparison of lift-curve slopes between different spanwise positions and the aerofoil. Here $\alpha _{{eff}}$ is determined by matching the 3-D lift coefficient at each spanwise section to the corresponding 2-D aerofoil lift coefficient.

Aerodynamic analysis reveals that the variation in lift-curve slope is driven by downwash flow from the root to the middle of the blade. To identify the physical mechanism responsible for the spanwise variation of the lift-curve slope, the lift coefficients of the 3-D blade sections with those of the 2-D S809 aerofoil are compared. As shown in figure 6, for the lift coefficient of various cross-sections along the 3-D blade at a free-stream angle of attack $\alpha _{{fs}}$ , an effective angle of attack $\alpha _{{eff}}$ exists, and its corresponding lift coefficient matches the 3-D value. Figure 6 thus serves as the phenomenological basis for the downwash model. Consequently, the 3-D aerodynamic force can be represented as

(2.4) \begin{align} C_{\!L}(\alpha _{{fs}}) = C_{l}(\alpha _{{eff}}). \end{align}

Figure 7.

Comparison of pressure distributions between the CFD results of the 2-D aerofoil and 3-D sections at 30 % spanwise positions of the standstill blade under different actual angles of attack: (a) $\alpha _{{fs}}$  = −6.1 $^\circ$ , (b) $\alpha _{{fs}}$  = −1.1 $^\circ$ , (c) $\alpha _{{fs}}$  = 3.9 $^\circ$ .

Figure 7. Long description

Three line graphs compare pressure distributions between 2-D aerofoil and 3-D sections at different angles of attack. Panel A: The line graph shows pressure coefficient (C_p) versus x/c for angles of attack −6.1 degrees. Panel B: The line graph shows pressure coefficient (C_p) versus x/c for angles of attack −1.1 degrees. Panel C: The line graph shows pressure coefficient (C_p) versus x/c for angles of attack 3.9 degrees. Each panel includes three data series: 2-D alpha with red circles, 2-D alpha with blue circles, and 3-D alpha with black squares. The x-axis represents the normalized chord length (x/c) ranging from 0 to 1, and the y-axis represents the pressure coefficient (C_p) with varying ranges.

The lift-curve slope predicted by the fitting function, together with the formulation of (2.4), is employed to compare the 2-D and 3-D results at $\alpha _{{fs}}$ and its corresponding $\alpha _{{eff}}$ , respectively. Figure 7 presents a comparison between the 2-D pressure distribution (solid red line) and the 3-D pressure distribution (black dots) at the 30 % spanwise section. The results show a marked difference between the 2-D and 3-D flow characteristics. The blue line represents the pressure distribution corresponding to the $\alpha _{{eff}}$ experienced by the blade sections under this operating condition. The comparison reveals that the 3-D pressure distribution at $\alpha _{{fs}}$ aligns closely with the 2-D pressure distribution at $\alpha _{{eff}}$ . It further indicates that the consistent lift coefficient shown in figure 6 is rooted in the matching behaviour of the pressure coefficient under attached-flow conditions. Consequently, the pure downwash effect merely induces an alteration in the angle of attack. The sections from the 30 % spanwise position to the 80 % spanwise position satisfy the principle of equivalent angle of attack.

Figure 8.

Comparison of pressure distributions between the CFD results of the 2-D aerofoil and 3-D sections at 47 % spanwise position of the rotated blade under different actual angles of attack: (a) $U_0$  = 5.0 m s⁻¹, (b) $U_0$  = 6.0 m s⁻¹, (c) $U_0$  = 7.0 m s⁻¹.

In figure 8, consistent with the non-rotating blade, the simulation results on the rotating blade always match those of the 2-D aerofoil at the corresponding $\alpha _{{eff}}$ under attached flow. As demonstrated in (2.1), the free-stream angle $\alpha _{{fs}}$ is determined by the interplay between the rotational speed and the free-stream velocity $U_{0}$ . Consequently, as the wind speed increases from 5.0 m s⁻¹ to 7.0 m s⁻¹, the free-stream angle of attack at the 47 % spanwise location increases from $4.9^\circ$ to $9.2^\circ$ . Moreover, the difference between $\alpha _{{fs}}$ in the 3-D flow and $\alpha _{{eff}}$ widens with wind speed, increasing from $0.9^\circ$ to $1.7^\circ$ when the wind speed rises from 5.0 to 7.0 m s⁻¹. This trend thereby confirms that the downwash effect becomes more pronounced at higher angles of attack.

Figure 9.

Pressure distribution on the suction surface of a stationary blade at different pitch angles, along with the corresponding free-stream angle of attack from 30 % to 95 % spanwise locations with different pitch angles: (a) $\beta$  = 81 $^\circ$ ,(b) $\beta$  = 71 $^\circ$ .

Figure 9. Long description

Panel A: A heat map showing pressure distribution on the suction surface of a stationary blade at different pitch angles. The x-axis represents the spanwise locations from 30 percent to 95 percent, and the y-axis represents the pressure coefficient (C_p). The color scale ranges from −1.5 to 1.5, with blue indicating lower values and red indicating higher values. Specific pitch angles are labeled at different spanwise locations: 10.6 degrees at 95 percent, 9.4 degrees at 80 percent, 7.9 degrees at 63 percent, 4.5 degrees at 47 percent, and −5.2 degrees at 30 percent. Panel B: Another heat map showing pressure distribution on the suction surface of a stationary blade at different pitch angles. The x-axis represents the spanwise locations from 30 percent to 95 percent, and the y-axis represents the pressure coefficient (C_p). The color scale ranges from −1.5 to 1.5, with blue indicating lower values and red indicating higher values. Specific pitch angles are labeled at different spanwise locations: 20.6 degrees at 95 percent, 19.4 degrees at 80 percent, 17.9 degrees at 63 percent, 14.5 degrees at 47 percent, and 5.2 degrees at 30 percent.

To further investigate the downwash effect of adjacent cross-sections, the upper-surface pressure distributions of the stationary blade at two different pitch angles are illustrated in figure 9. Here $\alpha _{{fs}}$ increases from root to tip, creating a spanwise pressure gradient near the leading edge, while a weaker gradient persists near the trailing edge. The cross-sectional aerodynamics exhibit quasi-2-D characteristics under the pressure-gradient conditions illustrated in figure 9(a). In addition, in figure 9(a) the equivalent-angle-of-attack principle of the pressure distributions at sections from a 80 % to 30 % span are consistent with the typical results shown in figure 7, i.e. that the 3-D pressure distribution collapses onto its 2-D counterpart at $\alpha _{{eff}}$ . Although flow separation becomes pronounced on outboard sections in figure 9(b), the inboard region near the 30 % span maintains attached flow, and its behaviour remains consistent with the effective angle of attack criterion. Thus, the spanwise distribution of the angle of attack shapes the spanwise flow structure, which in turn determines the local $\alpha _{{eff}}$ .

Figure 10 presents the upper-surface pressure distributions of the blade rotating at 72 rpm. Subfigures (a) and (b) correspond to incoming wind speeds of 7.0 and 9.0 m s⁻¹, respectively. Due to the velocity component induced by rotation, $\alpha _{{fs}}$ is larger near the root than at the tip. From the root to the mid-span, the modest gradient in $\alpha _{{fs}}$ produces only a weak spanwise pressure gradient on the upper surface. Furthermore, the rotation reduces the spanwise gradient between root and tip relative to the stationary blade.

Figure 10.

Pressure distribution on the suction surface of a rotational blade at different free-stream velocities: (a) $U_{0}$  = 7.0m s⁻¹, (b) $U_{0}$  = 9.0 m s⁻¹.

In summary, under attached-flow conditions, the downwash effect ensures that the pressure distribution on a 3-D blade section collapses onto the corresponding 2-D aerofoil pressure at a specific angle of attack. For the attached-flow conditions examined here, the pressure distribution at a given spanwise section can be predicted from the 2-D aerodynamics data evaluated at the corresponding $\alpha _{{eff}}$ . For both parked and rotating blades, the downwash effect is driven by the spanwise pressure gradient and is physically expressed as a modification of the local $\alpha _{{eff}}$ . Consequently, within the attached-flow regime considered here, the primary effect of rotation on the downwash is captured through the adjustment of the local $\alpha _{{fs}}$ , while the underlying downwash physics remains unchanged.

The preceding analysis indicates that the downwash exhibits a distinct quasi-2-D and localised character. Consequently, modelling the downwash effect can be approached by directly relating changes in inflow conditions and geometry, thereby mapping the 3-D aerodynamic data from 2-D aerofoil characteristics. The $\alpha _{{eff}}$ can be presented as

(2.5) \begin{align} \alpha _{{eff}} = f_{{downwash}}(\underbrace {r/R,c/R,\ \ldots }_{\text{geometry}},\underbrace {\alpha _{{fs}},U_{0},\ \ldots }_{\text{flow}}) .\end{align}

Based on the above analysis, a physical model of the downwash effect is established, as shown in figure 11.

Figure 11.

The diagram of downwash effect and the effective angle $\alpha _{{eff}}$ with the corresponding axial velocity $\Delta U$ .

To account for the downwash effect while remaining consistent with classical lifting-line theory, the 2-D sectional angle of attack is defined as $\alpha _{\text{2-D}} = \alpha _{{fs}} - \alpha _{0}$ , with $\alpha _{0}$ being the zero-lift angle of attack of the corresponding aerofoil. This definition serves not only to normalise the angle of attack but also to incorporate aerodynamic twist into the model.

Since the downwash effect primarily influences the velocity component normal to the aerofoil chord (denoted as $U_{0y}$ ), a velocity deficit $\Delta U$ is introduced in this normal direction to represent the downwash-induced variation. Meanwhile, the tangential velocity component parallel to the chord ( $U_{0x}$ ) is assumed to remain unchanged. Consequently, $\alpha _{{eff}}$ accounting for axial downwash velocity variation is derived from the velocity triangle as

(2.6) \begin{align} \alpha _{{eff}} = \mathrm{arctan}\left(\frac {U_{0y}-\Delta U}{U_{0x}}\right)=\mathrm{arctan}\left(\frac {U_{0} \mathrm{sin}\alpha _{\text{2-D}}-\Delta U}{U_{0} \mathrm{cos}\alpha _{\text{2-D}}}\right)\!. \end{align}

The downwash velocity deficit $\Delta U$ in the equation is an unknown parameter to be determined. Hence, to facilitate the modelling process, an axial downwash velocity factor $F_{a}$ is defined in (2.7):

(2.7) \begin{align} F_{a}=\frac {\Delta U}{U_0} .\end{align}

Therefore, the downwash effect can be described by

(2.8) \begin{align} \alpha _{{eff}} = \mathrm{arctan}\left(\mathrm{tan}\alpha _{\text{2-D}}-\frac {F_a}{\mathrm{cos} \alpha _{\text{2-D}}}\right)\!. \end{align}

Given that the velocity change induced by downwash is small, $F_{a}$ is a dimensionless quantity less than unity. Within the downwash modelling framework, the subsequent sections will employ SR to model this velocity factor $F_{a}$ .

2.4. Tip-loss effect and the radial transport velocity model

Based on the established understanding of the downwash effect, this section investigates the mechanism for the significant drop in lift-curve slope observed in the tip region. Figure 12 presents the lift-curve slope at the 95 % span location (red solid line), the aerofoil lift-curve slope (black solid line) and the prediction from the downwash fitting function (blue dashed line). Under the free-stream angle of attack $\alpha _{{fs}}$ , the reduction in lift-curve slope leads to a deviation in lift coefficient $\Delta C_{\!L}$ between the actual value and the prediction of the fit function. Consequently, the 3-D lift coefficient at the tip can be expressed as

(2.9) \begin{align} C_{\!L}(\alpha _{{fs}}) = C_{l}(\alpha _{{eff}})+\Delta C_{\!L}(\alpha _{{eff}}) ,\end{align}

where $\alpha _{{eff}}$ is obtained with the axial downwash velocity model proposed in § 2.3. Thus, the key to modelling tip loss lies in the load deviation $\Delta C_{\!L}$ at different angles of attack.

Figure 12.

Comparison of lift-curve slopes between the fit function prediction, the 95 % spanwise position and the aerofoil. The aerodynamic lift near the tip region is not only determined by the downwash $\alpha _{{eff}}$ but also by the load deflection $\Delta C_{\!L}$ .

Figure 13.

Comparison of pressure distributions between the CFD results of the 2-D aerofoil and the sections at 95 % spanwise position of the standstill blade under different actual angles of attack: (a) $\alpha _{{fs}}$  = −0.3 $^\circ$ , (b) $\alpha _{{fs}}$  = −1.1 $^\circ$ , (c) $\alpha _{{fs}}$  = 9.6 $^\circ$ .

Figure 13. Long description

Panel A: A line graph shows pressure distributions for angles of attack at −0.3 degrees. The x-axis represents the normalized chordwise position (x/c) ranging from 0 to 1, and the y-axis represents the pressure coefficient (C_p) ranging from −1 to 1. Three data series are plotted: 2-D at −0.3 degrees (red line with circles), 2-D at −0.5 degrees (blue dashed line with circles), and 3-D at −0.3 degrees (black line with squares). Panel B: A line graph shows pressure distributions for angles of attack at 1.1 degrees. The x-axis represents the normalized chordwise position (x/c) ranging from 0 to 1, and the y-axis represents the pressure coefficient (C_p) ranging from −1.5 to 1. Three data series are plotted: 2-D at −1.1 degrees (red line with circles), 2-D at 2.0 degrees (blue dashed line with circles), and 3-D at −1.1 degrees (black line with squares). Panel C: A line graph shows pressure distributions for angles of attack at 9.6 degrees. The x-axis represents the normalized chordwise position (x/c) ranging from 0 to 1, and the y-axis represents the pressure coefficient (C_p) ranging from −4 to 1. Three data series are plotted: 2-D at 9.6 degrees (red line with circles), 2-D at 6.9 degrees (blue dashed line with circles), and 3-D at 9.6 degrees (black line with squares).

Figure 13 compares the pressure distributions at the 95 % spanwise location. Downwash is still present near the tip, leading to notable differences between the 2-D and 3-D pressure distributions at the same $\alpha _{{fs}}$ . Moreover, it can be observed that under $\alpha _{{eff}}$ , the suction-side pressure in the 3-D distribution is significantly weakened, whereas differences on the pressure side are minor. This discrepancy between the pressure coefficient of the aerofoil at $\alpha _{{eff}}$ and that of the 3-D blade section at the corresponding $\alpha _{{fs}}$ is defined as $\Delta C_{\!p}$ . This indicates that the quasi-2-D flow pattern induced by downwash is disrupted by a suction-side pressure deficit. Consequently, near-tip load loss prevents the lift slope from increasing continuously from root to tip. Moreover, this aerodynamic loss at the tip becomes more pronounced as the angle of attack rises.

The rotating blade exhibits the same behaviour as the stationary blade for blade elements near the tip, as shown in figure 14. The pressure coefficient on the 3-D section shows a clear $\Delta C_{\!p}$ from the 2-D aerofoil distribution at the corresponding $\alpha _{{eff}}$ , with the discrepancy primarily appearing on the suction side. The pressure loss on the suction side is distributed in an approximately uniform manner, which reduces the extent of the negative-pressure region and consequently lowers the aerodynamic load.

Figure 14.

Comparison of pressure distributions between the CFD results of the 2-D aerofoil and 3-D sections at 95 % spanwise position of the rotated blade under different actual angles of attack: (a) $U_0$  = 5.0 m s⁻¹, (b) $U_0$  = 6.0 m s⁻¹, (c) $U_0$  = 8.0 m s⁻¹.

To further correlate tip load loss with fluid characteristics and clarify the flow-loading mechanism, the near-tip flow is examined in detail.

Figure 15 presents the streamwise ( $x$ -direction) velocity $u/U_{0}$ distribution around the stationary blade at different pitch angles. Due to flow acceleration over the suction side, the dimensionless velocity $u/U_{0}$ on the upper surface can exceed unity. In the mid-span region, the velocity isolines near the suction surface are parallel to the blade surface. However, approaching the tip, a pronounced velocity deficit is observed. This zone also exhibits strong spanwise transport.

Figure 15.

Normalised streamwise velocity $u/U_{0}$ distribution under different pitch angles. A pronounced velocity reduction is observed on the suction side near the tip, while the velocity on the pressure side shows little variation. Results are shown for (a) $\beta$ $= 86^\circ$ and (b) $\beta$ $= 81^\circ$ .

A further comparison is conducted on the velocity fields at the 95 % span section among the stationary blade, the rotating blade and the 2-D aerofoil at $\alpha _{{eff}}$ ( $10.6^\circ$ ). The $\alpha _{{fs}}$ for both the rotating and stationary blades at the 95 % span section are $13.6^\circ$ . In figure 16(a) the grey dashed line delineates the contour of $u/U_0 = 1.4$ . This region is notably diminished near the tip of the 3-D blade compared with that of the aerofoil at $\alpha _{{eff}}$ . Due to the presence of the tip vortex, the velocity $u$ along the aerofoil surface experiences a substantial reduction. This decrease in streamwise velocity correlates with the reduction in the pressure coefficient depicted in figure 16(c). For the rotating blade, the centrifugal effect introduces additional energy into the flow. Consequently, the tip region demonstrates a superposition of tip loss and rotational augmentation effects. In figure 16(b) the dashed line represents the contour of $v/U_0 = 0.2$ . The results reveal that the spanwise velocity $v$ on the 3-D blade is greater, which is attributed to the vortex roll-up phenomenon.

Figure 16.

Comparison of flow fields at the 95 % cross-section of a stationary blade, a rotating blade and a 2-D aerofoil under $\alpha _{{eff}}$ : (a) streamwise normalised velocity $u/U_{0}$ , (b) normal normalised velocity $v/U_{0}$ , (c) pressure coefficient $C_p$ .

Figure 17.

Total pressure loss coefficient ${\rm d}C_{pt}$ in the blade wake at $\beta =86^\circ$ : (a) distribution along the wake, (b) $r/R=47\,\%$ , (c) $80\,\%$ , and (d) $95\,\%$ , comparing the 3-D blade section and the aerofoil at the equivalent angle of attack.

Figure 17. Long description

Panel A: A 3-D model of a blade with a color gradient representing the distribution of total pressure loss coefficient (dCpt) along the wake. The color bar ranges from −0.50 to 0.46. Panel B: A line graph comparing the total pressure loss coefficient (dCpt) for a 2-D airfoil under effective angle of attack (αeff) and a 3-D blade section at 47 percent under angle of attack (αfs). The x-axis represents dCpt, and the y-axis represents the normalized spanwise coordinate (y/c). Panel C: A line graph comparing the total pressure loss coefficient (dCpt) for a 2-D airfoil under effective angle of attack (αeff) and a 3-D blade section at 80 percent under angle of attack (αfs). The x-axis represents dCpt, and the y-axis represents the normalized spanwise coordinate (y/c). Panel D: A line graph comparing the total pressure loss coefficient (dCpt) for a 2-D airfoil under effective angle of attack (αeff) and a 3-D blade section at 95 percent under angle of attack (αfs). The x-axis represents dCpt, and the y-axis represents the normalized spanwise coordinate (y/c).

To further illustrate the effect of the blade tip vortex on the reduction in tip velocity, the total pressure loss coefficient ${\rm d}C_{{pt}}$ is introduced to quantify the energy loss in the 3-D blade tip wake. Here, ${\rm d}C_{{pt}}$ is defined as the difference between the local total pressure $P_{\mathit{0, local}}$ and the free-stream total pressure $P_{0,\infty }$ :

(2.10) \begin{align} {\rm d}C_{{pt}}=\frac {P_{0,\infty }-P_{\mathit{0, local}}}{\dfrac {1}{2}\rho U_{0}^2} .\end{align}

In figure 17(a) a region of high-energy dissipation is observed in the tip-vortex area. Figure 17(b–d) presents ${\rm d}C_{{pt}}$ along characteristic lines perpendicular to the blade section, ranging from the mid-span to the tip. These results are compared with the ${\rm d}C_{{pt}}$ in the wake of the aerofoil at $\alpha _{{eff}}$ for each corresponding section. The results show that, at $r/R=47\,\%$ in the mid-span region, the energy dissipation characteristics remain largely consistent with the equivalent-angle-of-attack effect. However, the value of ${\rm d}C_{{pt}}$ of the blade tip is lower than that of the aerofoil at $\alpha _{{eff}}$ (figures 17 c and 17 d). In addition, a high-energy region can be observed in the blade tip-vortex region. Therefore, the flow energy is not only dissipated by viscous effects but also transferred into the tip vortex near the blade tip. The energy loss in the tip region consists of both viscous dissipation and spanwise transport.

To quantify the tip-loss characteristics on the basis of the downwash effect, the energy loss coefficient $\psi$ is introduced as

(2.11) \begin{align} \psi = \int {\rm d}C_{{pt},\text{2-D}}(\alpha _{{eff}}) {\rm d}(y/c)-\int {\rm d}C_{{pt},\text{3-D}}(\alpha _{{fs}}) {\rm d}(y/c). \end{align}

This coefficient represents the difference in the integrated total ${\rm d}C_{{pt}}$ between the 2-D and 3-D cases. Consistent with the tip load reduction, the energy loss coefficient decreases progressively from the tip toward the root. As shown in figure 18, the energy loss coefficient at the 95 % spanwise section exceeds that at the 47 % section by approximately one order of magnitude.

The quantified coefficient $\psi$ of the tip-loss effect in the flow field is compared with the aerodynamic force reduction in figure 18. The reduction in tip load $\Delta C_L$ follows the same trend as the energy loss coefficient observed in the numerical simulations. This further implies that the 3-D energy dissipation quantified by $\psi$ and the load reduction $\Delta C_L$ are physically linked. However, BEMT completely neglects the tip loss of the blade element model, and consequently, its predicted $\Delta C_L$ is always zero. At the blade root, the energy loss induced by spanwise flow is relatively small, and the BEMT remains effective.

Therefore, the tip vortex reduces the flow energy in the tip region, thereby decreasing the near-surface velocity on the suction side and weakening the suction-side pressure.

Figure 18.

Total pressure loss coefficient $\psi$ in the blade wake and the tip-loss load $\Delta C_L$ at a pitch angle of $\beta = 86^\circ$ .

Figure 19.

Limiting streamlines of the blade surface and the Q-criterion isosurface ( $Q=3.0$ ) under various pitch angles: (a) $\beta$ $= 86^\circ$ , (b) $\beta$ $= 80^\circ$ .

Figure 19. Long description

Panel A: A heat map showing limit streamlines and Q-criterion isosurfaces on a blade surface at a specific pitch angle. The color scale ranges from −3.0 to 1.8, with darker colors indicating lower values and lighter colors indicating higher values. The streamlines and isosurfaces are depicted in various colors, with red lines highlighting specific regions. Panel B: A heat map showing limit streamlines and Q-criterion isosurfaces on a blade surface at a different pitch angle. The color scale is the same as in Panel A, ranging from −3.0 to 1.8. The streamlines and isosurfaces are depicted in various colors, with red lines highlighting specific regions.

Based on the above analysis, the characteristic of velocity reduction is further examined to support the theoretical model. The decrease in streamwise velocity along the blade surface arises from the interaction between the tip vortex and the limiting streamlines. Figure 19 illustrates the limiting streamlines on the blade surface and the Q-criterion isosurface for two distinct pitch angles. Owing to the spillage of the tip vortex and the induced downwash, the streamwise velocity on the suction side is reduced, and fluid momentum is transferred from the suction surface into the tip vortex. Consequently, under the influence of the tip-vortex perturbation, the limiting streamlines at these pitch angles are deflected towards the inboard region.

Finally, a coherent physical pathway is established: the tip vortex and downwash reduce the momentum and velocity on the suction surface, leading to streamline deflection and a consequent decrease in aerodynamic load.

Modelling the velocity deficit on the upper surface and linking it to the pressure coefficient enables the prediction of tip load loss. Consequently, the tip load loss can be expressed as a function of the geometry and the downwash characteristics. Within a specific range of geometries, the coupling between the tip load and the downwash flow can be represented by the following expression:

(2.12) \begin{align} \Delta C_{\!L} = g_{{tiploss}}(\underbrace {r/R,c/R,\ \ldots }_{\text{geometry}},\underbrace {\alpha _{{eff}},U_{0},\ \ldots }_{\text{flow}}) .\end{align}

Therefore, a tip-loss effect model is proposed as illustrated in figure 20. The figure compares the velocity and pressure distributions between a real blade tip and an infinite-span case, labelled as $U _{\text{3-D}}$ , $p _{\text{3-D}}$ and $U _{\text{2-D}}$ , $p _{\text{2-D}}$ , respectively. To characterise the pressure loss resulting from the suction-side pressure difference between the 2-D and 3-D flows, a radial velocity $U_{r}$ is introduced. Here $U_{r}$ is used to quantify the velocity deficit induced by tip leakage:

(2.13) \begin{align} U _{\text{3-D}} = U _{\text{2-D}} - U_{r} .\end{align}

The velocity loss on the upper surface dominates the load loss, and the spanwise flow on the lower surface is relatively weak, as shown in figure 15. Bernoulli’s theorem is employed to relate velocity to pressure. It is worth noting that the use of Bernoulli’s theorem substantially simplifies the tip aerodynamic model. Despite Bernoulli’s equation being founded on inviscid assumptions, the model incorporates an undetermined function to compensate for errors arising from these physical simplifications. The importance of using Bernoulli’s theorem as the physical guidance for the 3-DGAR model is discussed in detail in Appendix B2.

Figure 20.

Diagram of the tip-loss effect. Panels (a,b) show the 3-D pressure, velocity and radial velocity at the blade tip compared with the 2-D velocity and pressure for an infinite span under the same inflow.

For the tip section of the blade and the corresponding 2-D aerofoil under the same incompressible inflow $U_{0}$ , the total pressure $P_{0}$ is identical. Hence, the pressure on the upper surface can be written as

(2.14) \begin{align} p _{\text{2-D}}+\frac {1}{2}\rho U _{\text{2-D}}^{2}=p _{\text{3-D}}+\frac {1}{2}\rho U _{\text{3-D}}^{2} .\end{align}

A pressure deficit exists between $p _{\text{2-D}}$ and $p _{\text{3-D}}$ , which is modelled by introducing a radial flow velocity:

(2.15) \begin{align} p _{\text{2-D}}+\frac {1}{2}\rho U _{\text{2-D}}^{2}=p _{\text{3-D}}+\frac {1}{2}\rho (U _{\text{2-D}}-U_{r})^{2} .\end{align}

Since the velocity and pressure on the upper surface of the aerofoil are not constant but vary along the chord, the aerodynamic force must be obtained by integrating both the pressure and dynamic pressure terms over the chord. Furthermore, figures 13, 14 and 15 show that under the equivalent angle of attack, the pressure distributions of the lower surface of the 2-D and 3-D cases are nearly identical. Therefore, by adding the same lower surface integral value to both sides of the equation, the expression for the lift force can be derived:

(2.16) \begin{align} l+\int \frac {1}{2}\rho U _{\text{2-D}}^{2} \boldsymbol{\cdot }{\rm d}c=L+\int \frac {1}{2} \rho (U _{\text{2-D}}-U_{r})^{2}\boldsymbol{\cdot }{\rm d}c .\end{align}

Here, $l$ and $L$ represent the dimensional lift in two and three dimensions, respectively. Given that both the aerofoil lift coefficient $C_{l}$ and the blade section lift coefficient $C_{\!L}$ are available in the training dataset, the radial velocity $U_{r}$ must be identified from these data. To this end, a simplifying assumption is introduced: the upper-surface velocity distribution is approximated as uniform with magnitude $U_{0}$ . The errors $e_{1}$ and $e_{2}$ incurred on both sides of the equation under this assumption are collectively represented by introducing an auxiliary expression for $U_{r}^{*}$ . In other words, (2.18) is derived under a set of physical simplifications, and the deviations caused by these simplifications are absorbed into the undetermined term $U_{r}^{*}$ :

(2.17) \begin{align} l+\frac {1}{2}\rho U_{0}^{2} \boldsymbol{\cdot }c +e_{1} =L+\frac {1}{2} \rho (U_{0}-U_{r})^{2}\boldsymbol{\cdot }c+e_{2}, \end{align}

(2.18) \begin{align} l+\frac {1}{2}\rho U_{0}^{2} \boldsymbol{\cdot }c=L+\frac {1}{2} \rho (U_{0}-U_{r}^{*})^{2}\boldsymbol{\cdot }c. \end{align}

Therefore, the unknown term $U_{r}^{*}$ can be solved by the following equation:

(2.19) \begin{align} C_{\!L}=C_{l}+\frac {U_{r}^{*}(2U_{0}-U_{r}^{*})}{U_{0}^{2}} .\end{align}

Furthermore, the tip-loss effect can be described by (2.20):

(2.20) \begin{align} \Delta C_{\!L}=\frac {U_{r}^{*}(2U_{0}-U_{r}^{*})}{U_{0}^{2}} .\end{align}

To reduce the complexity of SR, a dimensionless radial velocity factor $F_{r}$ is introduced in (2.21):

(2.21) \begin{align} F_{r}=\frac {U_{r}^{*}}{U_{0} }. \end{align}

It should be noted that the available experimental data for $C_{l}$ and $C_{\!L}$ are obtained in real viscous flow, whereas the engineering model relies on the inviscid assumption of Bernoulli’s equation and the assumption of chordwise uniform velocity. The errors introduced by these idealisations are ultimately absorbed into the expression for $F_{r}$ . Although these physical simplifications sacrifice some accuracy in the derivation, the predictive fidelity of the model is restored through the dimensionless definition of $F_{r}$ and its subsequent identification via the SR method. Therefore, the load reduction caused by the tip-loss effect is expressed as

(2.22) \begin{align} \Delta C_{\!L}=2F_r-F_r^2. \end{align}

Therefore, the total expressions of the 3-DGAR model can be expressed by

(2.23) \begin{align} C_L(\alpha _{{fs}})= C_l\left( \arctan \left( \tan \alpha _{\text{2-D}} - \frac {F_a}{\cos \alpha _{\text{2-D}}} \right)\right) + 2F_r - F_r^2, \end{align}

(2.24) \begin{align} C_D(\alpha _{{fs}})= C_d\left( \arctan \left( \tan \alpha _{\text{2-D}} - \frac {F_a}{\cos \alpha _{\text{2-D}}} \right)\right) + C_{Di} .\end{align}

The mechanistic analysis and physical model presented above are valid for attached flows on both stationary and rotating blades, demonstrating their unified applicability. In summary, the 3-DGAR framework applies to both stationary blades and rotating blades.

Table 1.

The training and generalisation experimental data.

Table 1. Long description

A table detailing experimental data for training and generalisation of blade performance. The table has four rows and three columns. Column headers are Training, Span location generalisation, and Generalisation. Row labels are blank. Row 1: Training, Span location generalisation, NREL Phase VI Parked Blade: C_L r/R = 30 percent, 63 percent, 95 percent (S809). Row 2: Span location generalisation, NREL phase VI parked blade: C_L r/R = 30 percent, 63 percent, 95 percent (S809). Row 3: Generalisation, NREL Phase VI parked blade: C_D r/R = 30 percent, 47 percent, 63 percent, 80 percent, 95 percent (S809). Row 4: Generalisation, MEXICO rotor parked blade: C_L and C_D r/R = 25 percent, 35 percent (DU97-250); r/R = 60 percent (RISO-A121); r/R = 82 percent, 92 percent (NACA64418). Row 5: Generalisation, Geometric generalisation, Modern large-scale parked blade: C_L and C_D r/R = 40.8 percent (DU97-300). Row 6: Generalisation, Operation generalisation, NREL phase VI rotated blade: C_L r/R = 30 percent, 47 percent, 63 percent, 80 percent, 95 percent (S809).

3.1. Training and prediction data for 3-DGAR model

Some typical wind tunnel experiments are used in this study, as shown in figure 2. Furthermore, the comprehensive wind turbine experimental database ensures the universality and predictive accuracy of the 3-DGAR model. Based on the physical understanding of 3-D geometric effects, as little high-cost, high-precision wind tunnel test data as possible is used in the training process. As shown in table 1, only the lift coefficients corresponding to the attached flow at the 30 %, 63 % and 95 % span locations are utilised to train the 3-DGAR model. Since the downwash velocity model fundamentally predicts the disparity between $\alpha _{{fs}}$ and $\alpha _{{eff}}$ , employing solely aerodynamic coefficients from the linear region in the training dataset renders the model more exclusively focused on representing this phenomenon. Additionally, the drag coefficients $C_{D}$ are not used in the SR modelling process, yet our framework can still predict them. Rotated cases are further employed to test the model.

3.2. Closure terms identified by limited training data

Within the 3-DGAR framework, SR is applied to the experimental lift coefficient data of the NREL Phase VI blade under parked conditions (table 1) to derive explicit mathematical expressions for the two velocity factors: the axial factor $F_{a}$ and the radial factor $F_{r}$ . Since these dimensionless factors are not directly measurable, their corresponding values $F_{a}$ and $F_{r}$ in the training dataset are first derived using the model formulation from § 2. In this work, we employ the Python symbolic regression (PySR) package (Cranmer Reference Cranmer2023; Makke & Chawla Reference Makke and Chawla2024), which evolves expression trees through a multi-population genetic algorithm. The evolutionary procedure incorporates selection, crossover and mutation operations. This method also effectively identifies many equations in fluid mechanics (Wang et al. Reference Wang, Feng, Zhou, Chen, Liao and Chen2025; Yang et al. Reference Yang, Shan, Yang and Zhang2025; Zhang et al. Reference Zhang, Chen, Liao, Zhao, Xie, Liu and Lu2026b ), balancing physical interpretability and accuracy.

As shown in figure 21, the detailed workflow for identifying the expressions of the two velocity factors is presented. First, the axial velocity factor $F_{a}$ is identified based on experimental data from three spanwise sections of the NREL Phase VI blade under parked conditions. To enhance the physical interpretability of the input variables, the SR inputs are determined based on prior physical knowledge and flow mechanisms.

The input physical variables include the dimensionless chord length $c/R$ at each section, the dimensionless distance of the section from the blade root $r/R$ and the 2-D angle of attack $\alpha _{\text{2-D}}$ . Based on the mechanistic analysis detailed in § 2, both the downwash flow and the tip-loss streamline are significantly influenced by the local angle of attack. To integrate the geometric twist and the aerodynamic twist into the model, the variable $\alpha _{\text{2-D}}$ is introduced during the model derivation process, accounting for geometric twist, the inflow angle of attack and the zero-lift angle of attack.

The intensity of the spanwise flow also exhibits variation with respect to the radial position. The quantified tip-loss results in figure 17 demonstrate that the tip loss varies significantly with the spanwise location. Consequently, blade geometric parameters emerge as crucial inputs for the 3-DGAR model. Among these, the two most pivotal inputs are $c/R$ and $r/R$ , which collectively delineate the blade planform.

Additionally, aerofoil polar data are utilised as predictive inputs. The aerodynamic force of a specific aerofoil can be expressed as (3.1) and (3.2):

(3.1) \begin{align} C_l = C_l(\alpha _{{fs}}, Re, M_a), \end{align}

(3.2) \begin{align} C_d = C_d(\alpha _{{fs}}, Re, M_a). \end{align}

Figure 21.

The flow chart of the identification of the velocity factor $F_{a}$ and $F_{r}$ by PySR.

Figure 21. Long description

Panel A: The flowchart begins with input variables (c/R, r/R, alpha2-D) and input operators (addition, subtraction, multiplication, division, exponentiation, sine). It then proceeds through symbolic regression and fitting loss to determine the axial velocity factor (Fa) of the training data. This process iterates until convergence is achieved, resulting in the axial velocity factor (Fa) as a function of (c/R, r/R, alpha2-D). Panel B: The flowchart starts with input variables (c/R, r/R, alpha_eff) and input operators (addition, subtraction, multiplication, division, exponentiation, sine). It then proceeds through symbolic regression and fitting loss to determine the radial velocity factor (Fr) of the training data. This process iterates until convergence is achieved, resulting in the radial velocity factor (Fr) as a function of (c/R, r/R, alpha_eff).

Parameters that affect the polar data but do not fundamentally modify the 3-D flow characteristics (such as the Reynolds number) are not incorporated as direct inputs into the SR. In essence, without explicitly modelling transition or laminar flow, the influence of the Reynolds number on the aerodynamic characteristics is incorporated into the model through the polar curves.

Furthermore, both the mechanistic analysis and the model derivation are based on incompressible 3-D flow, a condition representative of normal blade operation. For larger rotors, compressibility effects in the tip region become non-negligible and would require additional correction models within the BEMT framework. Therefore, the Mach number is excluded as an input to the SR.

The function set involves addition, subtraction, multiplication, division, exponentiation, sine function, quadratic and cubic. At the $i$ th iteration, the identified function is expressed as $f_{i}(c/R, r/R, \alpha _{\text{2-D}})$ . The loss function is defined as the deviation between $f_{i}$ and the training data:

(3.3) \begin{align} \textit{Loss}_{{pre}}=\sum _{n=1}^{N} |F_{a}-f_{i}(c/R, r/R, \alpha _{\text{2-D}})|. \end{align}

The algorithm incorporates a complexity penalty term. The total loss of an expression comprises two parts: the predictive loss and the complexity penalty. The complexity penalty is implemented by multiplying the loss function by the term $\text{exp}(\text{frecency}[C(E)])$ , where $E$ represents the mathematical expression obtained by the SR. Here, the complexity $C$ of an expression includes the number of variables, constants and operations. In this context, the ‘frecency’ of $C$ refers to a combined measure of the recency of expressions with complexity $C$ that appear in the population:

(3.4) \begin{align} Loss=\textit{Loss}_{{pre}} \boldsymbol{\cdot }\text{exp}(\text{frecency}[C(E)]). \end{align}

The tree-structured expressions and their constants are optimised iteratively by minimising a loss function until convergence is achieved. From the resulting Pareto front, the expression that offers the best balance between complexity and predictive accuracy is selected. This process yields the final functional form of $F_{a}$ . After modelling the downwash effect over the entire blade span, the correction for tip loss and the symbolic expression for the radial velocity factor $F_{r}$ is obtained. The results of PySR and their corresponding errors and complexities are detailed in Appendix B. For additional details, the reader is referred to Cranmer (Reference Cranmer2023).

Consequently, after repeated validation tests, the following expression is selected as an optimal balance between accuracy and complexity:

(3.5) \begin{align} F_{a}=\frac {c_{1}\boldsymbol{\cdot }\alpha _{\text{2-D}}+c_{2}} {\dfrac {r}{R} \boldsymbol{\cdot }\dfrac {R}{c} }, \end{align}

(3.6) \begin{align} F_{r}=\frac {\alpha _{{eff}}} {c_{3} \boldsymbol{\cdot }\big(c_{4}-\mathrm{e}^{\frac {1}{r/R}}\big)} .\end{align}

Here, $c_{1}$ , $c_{2}$ , $c_{3}$ and $c_{4}$ are empirical parameters in the model. The values are automatically identified during the SR process as $-2.10$ , $0.20$ , $20.0$ and $2.70$ , respectively.

Therefore, the total expressions of the 3-DGAR model in (2.23) can be expressed by

(3.7) \begin{align} C_L(\alpha _{{fs}}) &= \frac {\partial C_l}{\partial \alpha }\boldsymbol{\cdot }\!\left ( \arctan \!\left ( \tan \alpha _{\text{2-D}} - \frac {-2.1\boldsymbol{\cdot }\alpha _{\text{2-D}}+0.2}{r/c \boldsymbol{\cdot }\cos \alpha _{\text{2-D}}} \right ) \right ) \nonumber\\[5pt] &\quad + 1 - \left [ 1 - \frac {\arctan \!\left ( \tan \alpha _{\text{2-D}} - \dfrac {-2.1\alpha _{\text{2-D}}+0.2}{r/c \boldsymbol{\cdot }\cos \alpha _{\text{2-D}}} \right )} {20\boldsymbol{\cdot }(2.7-\mathrm{e}^{R/r})} \right ]^2 .\end{align}

Regarding the induced drag due to downwash, classical theory states that (Anderson Reference Anderson2011)

(3.8) \begin{align} C_{{Di}}=C_{\!L} \boldsymbol{\cdot }\alpha _{i}=C_{\!L} \boldsymbol{\cdot }(\alpha _{{fs}}-\alpha _{{eff}}). \end{align}

Therefore, the drag coefficient in (2.24) can be expressed as

(3.9) \begin{align} C_D(\alpha _{{fs}})= \frac {\partial C_d}{\partial \alpha }\boldsymbol{\cdot }\!\left ( \arctan \!\left ( \tan \alpha _{\text{2-D}} - \frac {-2.1\boldsymbol{\cdot }\alpha _{\text{2-D}}+0.2}{r/c \boldsymbol{\cdot }\cos \alpha _{\text{2-D}}} \right ) \right )+C_{\!L}(\alpha _{{fs}}) \boldsymbol{\cdot }(\alpha _{{fs}}-\alpha _{{eff}}) .\end{align}

To obtain a compact analytical formulation, (3.7) replaces the interpolation of aerofoil lift coefficient data with the derivative term of the lift coefficient, thereby expressing the sectional lift directly as a function of $\alpha _{{eff}}$ . In practice, achieving accurate nonlinear predictions requires interpolating 2-D aerofoil force data.

3.3. Physical consistency of $F_a$ and $F_r$

Figure 22.

The trend of $F_{a}$ is shown against its governing parameters. (a) As the $\alpha _{\text{2-D}}$ rises from 4 $^\circ$ to 12 $^\circ$ , $F_{a}$ decreases from an initially large value at the root, while remaining zero at the tip. (b) At a given angle of attack, $F_{a}$ decreases with chord length, indicating a lower local aspect ratio and a more pronounced downwash effect.

Spanwise flow induced by geometric effects is associated with factors such as aspect ratio, aerodynamic twist and geometric twist. The $F_{a}$ expression identified from the experimental data reveals that the downwash effect scales inversely with the local blade element aspect ratio, which is defined as $r/c$ , presenting the ratio of the radial position to the local chord length. This parameter $r/c$ that dominates the downwash is similar to the parameter $c/r$ derived by Snel et al. (Reference Snel, Houwink and Bosschers1994) through dimensional analysis for characterising geometric effects. Notably, a similar mathematical structure appears in the rotationally induced 3-D correction model proposed by Bak et al. (Reference Bak, Johansen and Andersen2006), in which the rotation-enhanced pressure increment is given by (3.10):

(3.10) \begin{align} \Delta C_{\!p,\textit{rotated}} \propto \frac {1}{r/c}. \end{align}

Furthermore, the twist angle and the zero-lift angle of attack are incorporated into the variable $\alpha _{\text{2-D}}$ . Therefore, the twist effect is explicitly represented in the input. Thus, the proposed axial induction factor $F_{a}$ simultaneously accounts for geometric twist, aerodynamic twist and aspect ratio effects, which aligns with classical lifting-line theory.

Figure 22 compares the influence of different independent variables on the parameter $F_a$ . Figure 22(a) shows that at a fixed chord length, as the angle of attack increases, the value of $F_a$ in the root region transitions from positive to negative. Furthermore, figure 22(b) reveals that a reduction in chord length leads to a more pronounced downwash-induced change in the angle of attack, corroborating the relationship between the downwash effect and the local aspect ratio ( $r/c$ ) introduced in this study.

The radial velocity factor $F_{r}$ reveals a tip-loss variation described by an exponentially decaying function ( $1-\mathrm{e}^{R/r}$ ) from root to tip. Therefore, the values of $F_{r}$ increase toward the tip. The scope of influence exerted by the tip-loss effect ought to be restricted to the tip region. The present model follows the same exponential-decay form as the classical Prandtl–Glauert tip model (Glauert Reference Glauert1935), expressed as $\mathrm{e}^{-R/r}$ .

Furthermore, since the tip-loss factor $F_r$ is built upon the downwash model and the tip-loss flow is more significantly influenced by the spanwise location, the chord length does not appear directly in the expression for $F_r$ . Instead, the effect of the local aspect ratio is implicitly embedded in $\alpha _{{eff}}$ .

As the angle of attack rises, the tip-loss effect becomes more pronounced, accompanied by a growing radial flow component. Figure 23 shows that as $\alpha _{{eff}}$ increases, the value of $F_{r}$ progressively decreases in the tip region. This trend indicates that a higher angle of attack strengthens the spanwise flow near the tip. It is worth noting that previous mechanistic studies (Zhang et al. Reference Zhang, Hayostek, Amitay, He, Theofilis and Taira2020) have established a clear dependence of wingtip vortex behaviour on both the angle of attack and aspect ratio, which is correctly captured by the formulation of $F_r$ .

The quadratic term in (2.22) introduces a saturation effect that embodies a physical limit in the mathematical formulation. It ensures that as the radial velocity $U_r$ approaches the free-stream velocity $U_{\mathrm{2-D}}$ , the growth rate of the dynamic pressure deficit gradually decreases and eventually converges to the free-stream dynamic pressure itself. It should be noted that this regime corresponds to significantly separated flow conditions, where the aerodynamic behaviour is no longer governed primarily by downwash or tip-loss mechanisms. Moreover, the quadratic dependence from Bernoulli’s theorem ensures that $F_r$ cannot increase unboundedly as the angle of attack increases. A detailed discussion of these results is provided in Appendix B.

Comprehensively, the combined effects of the $F_{a}$ and $F_{r}$ govern the evolution of the spanwise flow from the root to the tip. These factors bridge the geometric description of the blade to its aerodynamic loading output. Given the geometric and twist similarities common among HAWT blades, the applicability of the proposed 3-DGAR model will be further discussed in the following sections.

Figure 23.

The trend of the parameter $F_{r}$ as the $\alpha _{{eff}}$ increases from 4 $^\circ$ to 12 $^\circ$ . In the tip region, $F_{r}$ decreases from an initially higher value, whereas it remains zero at the blade root across all incidence angles.

4.1. Parked blade

First, testing is conducted for the non-rotating blade case. Under uniform and steady inflow, the 3-D loads of non-rotating blades are predicted by straightforward interpolation from 2-D polars within the framework of BEMT. This is achieved by integrating the inflow angle with the local twist angle at each blade section. As a result, in figures 24 and 25 the BEMT predictions exhibit identical values across all blade sections when plotted against the abscissa $\alpha _{{fs}}$ .

Figure 24.

Predicted lift coefficients of the NREL Phase VI with standstill condition. The prediction results of the 3-DGAR model and the conventional element model are presented with red solid lines and grey dashed lines, respectively. Results are shown for (a) $r/R$  = 30 %, (b) $r/R$  = 47 %, (c) $r/R$  = 63 %, (d) $r/R$  = 80 %, (e) $r/R$  = 95 %.

Figure 25.

Predicted drag coefficients of the NREL Phase VI with standstill condition. The prediction results of the 3-DGAR model and the conventional element model are presented with red solid lines and grey dashed lines, respectively. Results are shown for (a) $r/R$  = 30 %, (b) $r/R$  = 47 %, (c) $r/R$  = 63 %, (d) $r/R$  = 80 %, (e) $r/R$  = 95 %.

Figure 25. Long description

Five line graphs depict the drag coefficients (C_D) of the NREL Phase VI under standstill conditions. Each graph shows the relationship between the drag coefficient and the angle of attack (alpha_fs) in degrees. The spanwise positions are (a) 30 percent, (b) 47 percent, (c) 63 percent, (d) 80 percent, and (e) 95 percent. The x-axis represents the angle of attack (alpha_fs) in degrees, ranging from −10 to 30 degrees. The y-axis represents the drag coefficient (C_D), ranging from 0 to 0.3. Each graph includes three data series: a red solid line representing the present model, a grey dashed line representing the BEMT model, and black squares representing experimental data. The graphs show that the drag coefficient increases with the angle of attack, with notable differences between the present model, the BEMT model, and the experimental data.

As depicted in figure 24, in comparison with the classical BEMT method, the 3-D aerodynamic forces predicted by the present 3-DGAR model (the red solid line) exhibit good agreement with the experimental data (the black points) from the blade root to the tip. The present model accurately captures the changes in lift slope due to the downwash. Nevertheless, the lift-curve slope computed by the BEMT method remains identical to that of a 2-D aerofoil.

Given that the flow mechanisms and training data used in this study are derived from linear attached flows, it is somewhat unexpected that the 3-DGAR model still exhibits notable improvement in some stall conditions. As $\alpha _{{fs}}$ increases, the flow transitions into a complex 3-D nonlinear separated regime. Although the post-stall flow is coupled with complex separation and spanwise flow, the 3-DGAR model successfully captures the pure downwash effect dominated by blade geometry. However, it lacks the capability to predict 3-D separated flow effects, which still require additional corrections. A detailed discussion of the flow mechanisms underlying this behaviour is provided in § 4.4.

Figure 25 demonstrates that, although no drag data are included in the training dataset, the 3-DGAR model is still capable of accurately capturing the significant increase in 3-D drag. Correspondingly, the 3-DAGR model can be interpreted as essentially translating the 2-D drag coefficient and adding the lift-induced drag $C_{{Di}}$ that varies with $\alpha _{{eff}}$ . However, when the angle of attack far exceeds the static stall angle, the flow exhibits severe separation and unsteady effects. Consequently, some discrepancies arise in the predicted aerodynamic coefficients at high angles of attack. Nevertheless, the force coefficients corrected for geometric effects still show trends consistent with the experimental data as the angle of attack varies.

Additionally, the integrated loads of the NREL Phase VI blade are computed. Figure 26 illustrates a comparison of the predicted thrust $C_{{th}}$ and torque $C_{{tq}}$ coefficients for pitch angles spanning from 90 $^\circ$ to 70 $^\circ$ under stationary conditions. Due to the absence of experimental aerofoil data for angles of attack below −10 $^\circ$ , integrated load predictions are limited to pitch angles around 90 $^\circ$ . By incorporating 3-D geometric effects, the current model substantially enhances the accuracy of integrated blade load predictions.

Figure 26.

Predicted thrust $C_{{th}}$ and torque $C_{{tq}}$ coefficients for pitch angles spanning from 90 $^\circ$ to 70 $^\circ$ under stationary conditions: (a) $C_{{th}}$ , (b) $C_{{tq}}$ .

Figure 27.

Predicted lift and drag coefficients of the MEXICO with standstill condition. The prediction results of the 3-DGAR model and 3-DGA model (without a tip-loss factor) are presented with red solid lines and blue dashed lines, respectively. Results are shown for (a,d) $r/R$  = 25 %, (b,e) $r/R$  = 60 %, (c, f) $r/R$  = 95 %.

The 3-DGAR model is further validated using the MEXICO rotor, which features three distinct aerofoil profiles. To assess the influence of tip-loss effects at different spanwise positions, two models are compared: the full 3-DGAR model incorporating both downwash and tip-loss effects, and the 3-DGA model that considers only downwash effects. As shown in figure 27, for the DU97-250 aerofoil near the blade root, 3-DGAR predictions show excellent agreement with measurements. For the mid-span RISO-A121 and tip-section NACA64418 aerofoils, some discrepancies occur at high angles of attack due to differences in stall behaviour. The tip-loss factor reduces lift coefficients at high angles of attack for the 82 % and 92 % spanwise stations. The exponentially decaying expression in the 3-DGAR model successfully captures the decay of tip-loss effects with spanwise position. The 3-DGAR model also yields high predictive accuracy for additional cross-sections.

The drag coefficient of the MEXICO rotor is tested, as shown in figure 27(d–f). At low angles of attack, after accounting for induced drag and $\alpha _{{eff}}$ , the 3-D experimental results can be perfectly predicted. MEXICO rotor in figure 27(f) differs from that of the NREL Phase VI blade in figure 25(e): the former exhibits an increase in drag coefficient toward the tip, whereas the latter shows a marked reduction. Whether these two opposing physical phenomena stem from experimental measurement errors or indicate that other physical effects of the drag coefficient warrant further investigation. The integrated loads for this blade are summarised in table 2.

Figure 28 presents the predicted lift and drag at the 40.8 % span position of a large 87 m blade. This blade section utilises the thicker DU97-300 aerofoil, and its geometry is more representative of modern, large-scale wind turbines. Due to the higher local aspect ratio of this section, a slight reduction in the lift-curve slope is observed. For this test case, the uncorrected 2-D aerofoil polars (grey dashed line) already provide a reasonable prediction of the lift coefficient, particularly in the mid $\alpha _{{fs}}$ range.

Figure 28.

Predicted (a) lift and (b) drag coefficients of the large-scale blade with standstill condition of 40.8 %.

The limited improvement observed in the 3-DGAR model’s performance for this particular blade geometry can be attributed to the following factors. Firstly, the blade length and the 40.8 % spanwise location contribute to a relatively weak downwash effect, so the advantage of the 3-DGAR model is not pronounced (only a 4.8 % relative error improvement in the linear regime). Secondly, the scaled aerofoil wind tunnel data are employed in the 3-DGAR model for comparison with the simulation data of 3-D blades obtained from Cai et al. (Reference Cai, Yang, Jia, Wu, Zhang, Yuan, Qian and Li2023). Inconsistencies in data sources introduce uncertainty. Ultimately the proposed 3-DGAR model preserves its predictive accuracy for large-scale blades without experiencing degradation or failure.

In summary, the validation results consistently indicate that geometric effect corrections are important for improving the accuracy of load predictions under parked conditions, particularly near the blade root and tip with more pronounced spanwise flow. A detailed error comparison will be demonstrated in the subsequent sections.

4.2. Rotated blade

To validate its generalisability under rotating conditions, the 3-DGAR model is directly applied to modify the classical rotational enhancement model. The load augmentation of rotational effect and the restructuring of pressure distribution are described by the Bak model (Bak et al. Reference Bak, Johansen and Andersen2006), as shown in figure 4.

As shown in figure 29, significant downwash effects are also observed near the blade root and mid-span positions. Notably, at the 30 % spanwise location adjacent to the root as shown in figure 29(a), the variation in lift slope is remarkably significant. By correcting $\alpha _{{eff}}$ , the 3-DGAR-Bak model accurately captures both the change of the lift-curve slope and the rotational augmentation at high angles of attack. Moreover, the proposed tip-loss correction precisely predicts the reduction in aerodynamic forces at the 95 % span station under rotational effects. Furthermore, figure 29(e) confirms that the load loss becomes more pronounced at higher $\alpha _{{fs}}$ , which is consistent with the expression of the $F_{r}$ .

Figure 29.

The prediction of the NREL Phase VI under rotated conditions. The rotational enhancement is predicted by the Bak model (the blue dashed line). The rotational enhancement model with the downwash and tip-loss effect produces better precision (the red solid line). Results are shown for (a) $r/R$  = 30 %, (b) $r/R$  = 47 %, (c) $r/R$  = 63 %, (d) $r/R$  = 80 %, (e) $r/R$  = 95 %.

Figure 30.

Blade sectional loads predicted by the initial BEMT and those corrected by the 3-DGAR model for the NREL Phase VI blade: (a) $r/R=30\,\%$ , (b) $r/R=80\,\%$ , (c) $r/R=95\,\%$ .

Figure 30. Long description

Three line graphs compare blade sectional loads predicted by different models for the NREL Phase VI blade. Panel A: The line graph shows the relationship between the normal force coefficient (Cn) and the wind speed (U0) at 30 percent of the blade radius (r/R). The x-axis represents wind speed (U0) in meters per second (m/s), ranging from 5 to 25 m/s. The y-axis represents the normal force coefficient (Cn), ranging from 0 to 3. The graph includes three data series: Present (red dotted line with red circles), BEMT (blue solid line with blue circles), and Exp (black squares). Panel B: The line graph shows the relationship between the normal force coefficient (Cn) and the wind speed (U0) at 80 percent of the blade radius (r/R). The x-axis represents wind speed (U0) in meters per second (m/s), ranging from 5 to 25 m/s. The y-axis represents the normal force coefficient (Cn), ranging from 0.6 to 1.5. The graph includes three data series: Present (red dotted line with red circles), BEMT (blue solid line with blue circles), and Exp (black squares). Panel C: The line graph shows the relationship between the normal force coefficient (Cn) and the wind speed (U0) at 95 percent of the blade radius (r/R). The x-axis represents wind speed (U0) in meters per second (m/s), ranging from 5 to 25 m/s. The y-axis represents the normal force coefficient (Cn), ranging from 0.3 to 1.2. The graph includes three data series: Present (red dotted line with red circles), BEMT (blue solid line with blue circles), and Exp (black squares).

Additionally, the sectional loads predicted by the BEMT under rotating conditions are compared with experimental data across various wind speeds in figure 30. When augmented with the 3-DGAR correction, the BEMT framework yielded improved accuracy in aerodynamic load predictions. For the 95 % spanwise tip section, the combination of the proposed tip-loss correction and the classical tip induction velocity correction (Glauert Reference Glauert1935) effectively addressed the tip-load overprediction observed in figure 30(c). The relative errors of the BEMT method and the proposed method are also computed. At the blade tip, the BEMT method neglects the loss induced by geometric effects, resulting in a mean relative error of 93.57 %, whereas the proposed method yields a prediction error of only 19.69 %. At the 30 % section, the 3-DGAR model reduces the relative error by approximately 40 %.

The physical necessity of blade element correction lies in the fact that modifying the tip induced velocity alone cannot directly alter the tip load and lift-curve slope shown in figure 5. The induced velocity correction primarily reduces the sectional angle of attack, whereas changes in the lift-curve slope and loading require a direct correction to the blade element theory or the polar database. According to Boatto et al. (Reference Boatto, Bonnet, Avallone and Ragni2023), blade tip loss necessitates a correction to the 2-D lift coefficient to achieve more accurate predictions of the aerodynamic forces at the outboard blade section. Therefore, the tip-loss model developed in this study provides a physically based correction that complements the classical Prandtl tip-loss model for the momentum theory.

Without incorporating any experimental data from rotating blades in model training, the present 3-DGAR model successfully predicts the downwash effect and tip-loss effect on rotating blades.

4.3. Error analysis

Table 2 provides an error analysis for representative test cases conducted under both parked and rotating conditions. The relative error is employed to quantify the disparity between the predicted aerodynamic forces and the experimentally measured values. The original BEMT framework demonstrated notable prediction discrepancies in both the blade root and blade tip areas across all three case studies, whereas the prediction errors progressively diminished in the mid-span region of the blade. This further presents that spanwise flow effects are more prominent in the blade root and tip regions. The relative error analysis demonstrates that the 3-DGAR model yields a substantial enhancement in prediction accuracy for both lift and drag coefficients. ‘Reduc.’ denotes the percentage reduction in prediction error achieved by the present method relative to the baseline BEMT.

Table 2.

Comparison of relative errors (%) between the original BEMT and the improved BEMT corrected by the 3-DGAR model, relative to experimental ground truth.

Table 2. Long description

A table comparing relative errors between the original BEMT and the improved BEMT corrected by the 3-DGAR model. The table has five columns labeled Case, Force, Method, Section, and Reduc. It contains five rows of data for each of the three cases. Row 1: Case 1, Force Sectional, Method BEMT, Section 45.7 percent, Reduc. 18.7 percent. Row 2: Case 1, Force Sectional, Method Present, Section 68.8 percent, Reduc. 29.8 percent. Row 3: Case 1, Force Integrated, Method BEMT, Section 44.42 percent, Reduc. 20.1 percent. Row 4: Case 2, Force Sectional, Method BEMT, Section 19.2 percent, Reduc. 12.4 percent. Row 5: Case 2, Force Sectional, Method Present, Section 47.5 percent, Reduc. 25.7 percent. Row 6: Case 2, Force Integrated, Method BEMT, Section 24.48 percent, Reduc. 8.3 percent. Row 7: Case 3, Force Sectional, Method BEMT, Section 19.6 percent, Reduc. 20.7 percent. Row 8: Case 3, Force Sectional, Method Present, Section 57.6 percent, Reduc. 18.4 percent. Row 9: Case 3, Force Integrated, Method BEMT, Section 39.8 percent, Reduc. 24.9 percent. Row 10: Case 4, Force Sectional, Method BEMT, Section 12.7 percent, Reduc. 9.3 percent. Row 11: Case 4, Force Sectional, Method Present, Section 53.2 percent, Reduc. 20.6 percent. Row 12: Case 4, Force Integrated, Method BEMT, Section 28.3 percent, Reduc. 10.7 percent. Row 13: Case 5, Force Sectional, Method BEMT, Section 22.8 percent, Reduc. 12.7 percent. Row 14: Case 5, Force Sectional, Method Present, Section 48.9 percent, Reduc. 25.0 percent. Row 15: Case 5, Force Integrated, Method BEMT, Section 37.5 percent, Reduc. 62.2 percent.

Nevertheless, owing to the uncertainty inherent in the experimental drag coefficient near the 95 % span section of the MEXICO rotor, the relative error at this specific location surpasses 180 %. Excluding this particular data point results in a reduction of the prediction error for the drag coefficient by more than a factor of two (the values enclosed in parentheses for the MEXICO parked case in table 2 correspond to the error statistics derived when the 95 % section is excluded). Furthermore, the reduction in prediction error for the blade aerodynamic force ranges from 40 % to 60 %. These quantitative metrics further substantiate the efficacy of the 3-DGAR model.

Figure 31.

(a,b) Downwash velocity distributions $v_d$ on the upper and lower blade surfaces at (a) $\beta = 86^\circ$ and (b) $\beta = 76^\circ$ , superimposed with the $Q$ -criterion isosurface at $Q = 3.0$ . Corresponding predictions of the 3-DGAR model for (c) $F_a$ and (d) $F_r$ .

Figure 31. Long description

Panel A: A 3-D visualization of downwash velocity distributions on the upper surface of a blade. The color scale ranges from −0.13 to 0.15, with blue indicating lower velocities and red indicating higher velocities. Specific angles (alpha_fs) are marked at various points along the blade. Panel B: A 3-D visualization of downwash velocity distributions on the lower surface of a blade. The color scale ranges from −0.6 to 0.4, with blue indicating lower velocities and red indicating higher velocities. Specific angles (alpha_fs) are marked at various points along the blade. Panel C: A line graph showing the variation of the normal force (Fa) with respect to the normalized radial position (r/R). Two lines are plotted: a solid red line for beta = 86 degrees and a dashed blue line for beta = 76 degrees. Panel D: A line graph showing the variation of the tangential force (Fr) with respect to the normalized radial position (r/R). Two lines are plotted: a solid red line for beta = 86 degrees and a dashed blue line for beta = 76 degrees.

4.4. Discussion on model applicability

This section discusses the applicability of the 3-DGAR model from two perspectives: flow-physics consistency and geometric configuration.

4.4.1. Flow

As shown in figure 26, both BEMT and the 3-DGAR model perform well at low sectional angles of attack (pitch angles $\beta$ near 90 $^\circ$ ), especially for the lift-dominated integrated forces. As $\alpha _{{fs}}$ increases, both methods exhibit growing prediction errors, with BEMT failing significantly. This behaviour is evidently associated with the stall modes under different $\alpha _{{fs}}$ . Therefore, a comprehensive analysis combining quantified indicators of 3-D effects and flow structures is conducted to unveil the flow mechanisms underlying the aerodynamic predictions.

To reveal the linkage between the 3-DGAR model and the flow physics, the dimensionless downwash velocity $v_d$ is introduced as a quantitative measure of the downwash effect at different blade sections:

(4.1) \begin{align} v_d= \frac {v}{U_0} -\mathrm{sin}(90-\beta ) .\end{align}

Under small-angle attached-flow conditions, the $v_d$ distribution is dominated by the pure downwash effect, as shown in figure 31(a). At the blade root, the large geometric twist induces a strong downwash, whereas the tip exhibits an opposite downwash direction. Overall, the downwash velocity is distributed in a relatively uniform manner across the blade. When the pitch angle reaches $76^\circ$ in figure 31(b), flow separation occurs at the 50 % spanwise location. Toward the tip region, the separation becomes more pronounced as the $\alpha _{{fs}}$ continues to increase. At this stage, the downwash flow exhibits non-uniform characteristics near the tip. Near the 80 % spanwise position the more severe separation can be observed from the vorticity isosurface. The separation bubble in the tip region is weakened by the radial transport induced by the tip vortex (Zhu et al. Reference Zhu, Wang and Liu2024). This tip-flow weakening phenomenon is associated with the downwash effect, which inhibits boundary-layer separation near the wingtip (Toppings & Yarusevych Reference Toppings and Yarusevych2021). Consequently, the downwash velocity first decreases and then increases along the span.

As shown in figures 31(c) and 31(d), the 3-DGAR model successfully captures the growth of downwash velocity and tip loss (corresponding to $F_a$ and $F_r$ , respectively) with increasing $\alpha _{{fs}}$ . Evidently, consistent with the previous mechanistic analysis under attached flow, the 3-DGAR model does not capture the nonlinear characteristics induced by the coupling of separated flow and spanwise flow as shown in figure 31(b). Therefore, the 3-DGAR model can predict the variation of pure 3-D spanwise flow with $\alpha _{{fs}}$ , but its predictive capability is insufficient for strongly coupled flows involving separation bubbles.

Downwash velocities are then extracted at $0.5c$ downstream of the trailing edge on both the upper and lower surfaces. For the completely attached flow shown in figure 31(a), the predicted $\alpha _{{eff}}$ from the 3-DGAR model follows the same trend as the envelope of the upper and lower surface downwash velocities (figure 32 a). In terms of aerodynamic load prediction, BEMT exhibits significant errors for the 30 % span section (zone 1), which is reflected in the flow field as an underestimation of the downwash velocity. For the 80 % span section (zone 2), both BEMT and 3-DGAR yield results close to the measurements, as their predictions are consistent with the downwash velocities shown in figure 32(a).

Figure 32.

(a) Quantitative assessment of downwash predictions for the blade at $\beta = 86^\circ$ from the BEMT and 3-DGAR models; (b,c) corresponding load predictions at the 30 % and 80 % spanwise sections.

For the moderate separation around the stall angle occurring at the 63 % section (figure 31 a), the 3-DGAR model predictions remain within the envelope of the downwash velocities, as shown in figure 33(a). Although the 3-DGAR model is unable to capture the nonlinear 3-D flow induced by separation bubbles at high $\alpha _{{fs}}$ , it still yields better aerodynamic predictions than BEMT, as demonstrated in figure 33(b).

However, as flow separation intensifies, the 3-DGAR model struggles to capture the nonlinear variation in $v_d$ induced by the separation bubble. Nevertheless, it still yields more accurate predictions for the overall downwash distribution compared with BEMT, as shown in figure 33(a). With increasing $\alpha _{{fs}}$ , the prediction error of the 3-DGAR model becomes more pronounced. Under strong separated flow at high $\alpha _{{fs}}$ (figure 33 c, zone 2), additional corrections for 3-D separated flows are still required.

Figure 33.

(a) Quantitative assessment of downwash predictions for the blade at $\beta = 76^\circ$ from the BEMT and 3-DGAR models; (b,c) corresponding load predictions at the 63 % and 95 % spanwise sections.

Figure 33. Long description

Panel A: A line graph shows the quantitative assessment of downwash predictions for the blade from the BEMT and 3-DGAR models. The x-axis represents the normalized radial position (r/R) ranging from 0.3 to 0.9, and the y-axis on the left represents the effective angle of attack (α_eff) in degrees, while the y-axis on the right represents the lift coefficient (C_L). The graph includes data points for upper and lower downwash velocities from CFD and experimental lift coefficients. The BEMT model is represented by a dashed line, and the 3-DGAR model by a solid red line. The graph is divided into two zones: Zone 1 and Zone 2. Panel B: A line graph shows the corresponding load predictions at the 63% spanwise section. The x-axis represents the downwash velocity (v_d) in degrees, and the y-axis represents the lift coefficient (C_L). The BEMT model is represented by a dashed line, and the 3-DGAR model by a solid red line. Panel C: A line graph shows the corresponding load predictions at the 95% spanwise section. The x-axis represents the free stream angle of attack (α_fs) in degrees, and the y-axis represents the lift coefficient (C_L). The BEMT model is represented by a dashed line, and the 3-DGAR model by a solid red line.

Figure 34.

Variation of downwash velocity of the upper surface with pitch angle, along with the region of overlap between the predicted results from BEMT.

Furthermore, figure 24 shows that the BEMT predictions become notably less accurate at higher $\alpha _{{fs}}$ , which coincides with a marked increase in downwash velocity in figure 34. As a result, the effective range of BEMT progressively shrinks towards the blade root and eventually fails entirely.

In summary, the mechanism underlying the accurate predictions of the 3-DGAR model lies in its ability to capture the distributions of the downwash velocity. With increasing $\alpha _{{fs}}$ , the BEMT predictions of downwash velocity show progressively larger deviations from the real flow field. The 3-DGAR model, by contrast, provides more accurate predictions under attached-flow conditions, effectively improving the performance of the BEMT. Nevertheless, the nonlinear downwash behaviour arising from the interaction between separation bubbles and spanwise flow at high $\alpha _{{fs}}$ remains beyond the current model’s scope and calls for further theoretical development. Furthermore, this investigation is confined to subsonic flows, and the influence of compressibility on 3-D flow remains to be considered.

The 3-DGAR model provides a robust and physically consistent correction for spanwise flow effects in the attached and moderately separated regimes. For deep stall and compressible flows, the model offers partial improvements but requires further refinement. The clear correspondence between the correction factors and the underlying flow physics provides a foundation for future extensions to more complex flow regimes as shown in table 3.

Table 3.

Correspondence between the 3-DGAR correction factors and the underlying flow mechanisms.

Table 3. Long description

A table with three columns and four rows, comparing correction factors, symbolic expressions, and key geometric characteristics for different flow regimes. The columns are labeled Correction factor, Symbolic expression, and Key geometric characteristics. Row 1: Correction factor, F_a = f(alpha_2d, r/c); Symbolic expression, alpha_2d, r/c, Constants; Key geometric characteristics, alpha_0, alpha_fs, and gamma, c, r, c, gamma. Row 2: Correction factor, F_r = g(alpha_eff, r/R); Symbolic expression, alpha_eff, r/R, Constants; Key geometric characteristics, r, c, gamma.

4.4.2. Geometry

This study examines the geometric parameters of typical wind turbines with output power ranging from kW to 15 MW. Over the past two decades, substantial advancements in the design philosophy of HAWTs have translated into marked gains in power output. Nevertheless, as illustrated in figure 35, their normalised geometric parameters exhibit convergent trends, owing to characteristic aerodynamic behaviour coupled with multi-system design constraints. Due to the rotational, loading and structural characteristics of the blades, all the considered blades adopt positive twist at the root and negative twist at the tip, with twist angles falling within a similar range, approximately $15^\circ$ at the root and about $-2^\circ$ at the tip. Moreover, the chord length decreases gradually from the root to the tip.

Figure 35.

Distribution of the normalised twist angle and normalised chord length along the spanwise location for a representative wind turbine. All values are normalised by their respective maximum values.

The red points mark the experimental data used to validate the 3-DGAR model. Virtually all twist distributions of HAWTs fall within the valid range of the 3-DGAR model. The twist angle significantly governs the downwash velocity distribution through its effect on spanwise flow. Hence, for blades with comparable twist-angle distributions, a consistent downwash pattern can be expected (Pandi & Mittal Reference Pandi and Mittal2025). Based on the earlier mechanistic analysis and the derived velocity-factor expressions, these blades exhibit analogous geometric effects and downwash characteristics, given the similarity in their geometric parameters. Furthermore, these data provide important support for the applicability of the 3-DGAR model to aerodynamic predictions for typical wind turbines. Table 4 summarises the explicit mapping between the correction factors and the governing geometric parameters.

Table 4.

Correspondence between the 3-DGAR correction factors and the underlying geometric characteristics.

Table 4. Long description

A table with two columns: Complexity and Expression. The Complexity column lists values 1, 3, 4, 5, 7, 9, 11, 13, and 15. The Expression column contains corresponding mathematical expressions. Row 1: Complexity 1, Expression −0.00896. Row 2: Complexity 3, Expression −2.64791*α2-D. Row 3: Complexity 4, Expression −3.72875*(α2-D)^2. Row 4: Complexity 5, Expression −0.5943*α2-D/((r/R)/(c/R)). Row 5: Complexity 7, Expression −2.2453*α2-D/((r/R)/(c/R)). Row 6: Complexity 9, Expression −2.0979*α2-D − 0.1963/((r/R)/(c/R)). Row 7: Complexity 11, Expression −2.0979*α2-D − 0.1963/((r/R)/((c/R) − (0.08517))). Row 8: Complexity 13, Expression −2.0979*α2-D − 0.1963/((r/R)/((c/R) − (0.08517/0.9373))). Row 9: Complexity 15, Expression −2.0979*α2-D − 0.1963/((r/R)/(c/R) − (0.08512/(0.9476 − (r/R)))).

The 3-DGAR model exhibits a certain degree of applicability to configurations with similar geometric distributions. However, for high-aspect-ratio wings or untwisted straight wings, there exists a potential risk of failure.